On the automorphism group of a groupoid
type: Item
instance of: scholarly article
title: On the automorphism group of a groupoid
zbMATH Open document ID: 0933.20051
author: Domenico Freni
author: Nicola Rodinò
published in: Annales Mathématiques Blaise Pascal
publication date: 2000-01-17T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/79200
review text: For a groupoid \((X,\cdot)\) (in the sense of a set with a binary operation), the authors study its automorphism group as a permutation group of~\(X\) endowed with a family \((\sigma_i)_{i\in I}\) of transformations of \(X\) associated with the orbits \((X_i)_{i\in I}\) such that each \(\sigma_i\) commutes with the action of the stabilizer of \(X_i\). Conversely, they show how to associate with a faithful action of a group \(G\) on a set \(X\), together with a family of transformations of~\(X\) as above, a groupoid structure on~\(X\) in whose automorphism group \(G\) embeds naturally. Moreover, if the group was constructed as above from a groupoid, then the groupoid thus obtained is the original one. The paper goes on to analyze how various special properties on the groupoid relate to special properties of the group actions. In particular, they obtain a proof of a theorem of \textit{M. Gould} [Algebra Univers. 2, 54-56 (1972; Zbl 0248.20071)] stating that an at most countable group \(G\) has a left-cancellable groupoid structure whose automorphism group is isomorphic to~\(G\).
reviewed by: Jorge Almeida
Mathematics Subject Classification ID: 20N02
Mathematics Subject Classification ID: 20B27
Mathematics Subject Classification ID: 20F29
zbMATH DE Number: 1275893
zbMATH Keywords: groupoids
zbMATH Keywords: automorphism groups
zbMATH Keywords: group actions
MaRDI profile type: Publication
OpenAlex ID: W2086409406
cites work: Q5841992
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DOI: 10.5802/AMBP.108
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description: scientific article; zbMATH DE number 1275893